The 2-log-convexity of the Apéry numbers

Chen, William; Xia, Ernest

doi:10.1090/S0002-9939-2010-10575-0

The 2-log-convexity of the Apéry numbers
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by William Y. C. Chen and Ernest X. W. Xia PDF

Proc. Amer. Math. Soc. 139 (2011), 391-400 Request permission

Abstract:

We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apéry numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of orders $3$ and $4$ and the large Schröder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are $k$-log-convex for any $k\geq 1$ possibly except for a constant number of terms at the beginning.

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